{"paper":{"title":"Functions in $L_1(\\mu,Y)$ with optimal tensor representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Juan Guerrero-Viu, Luis C. Garc\\'ia-Lirola","submitted_at":"2026-07-02T14:50:42Z","abstract_excerpt":"We study the existence and characterization of optimal tensor representations of elements in the space $L_1(\\mu,Y)$ of Bochner integrable functions. We completely describe the set of norm-attaining elements in two settings. First, when the Banach space $Y$ is strictly convex, and second, when $Y=L_1(\\nu)$ and $\\mathbb K=\\mathbb R$. In both situations, our analysis yields the existence of non-norm-attaining tensors whenever the underlying measures are not purely atomic. Finally, we introduce a geometric property over $Y$ ensuring that every element in $L_1(\\mu, Y)$ admits an optimal representat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02263","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02263/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}